Commit 53b3b48cfde788f3f9debe40228b3edfc0f823ad
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1 | +## Core latex/pdflatex auxiliary files: | |
2 | +*.aux | |
3 | +*.lof | |
4 | +*.log | |
5 | +*.lot | |
6 | +*.fls | |
7 | +*.out | |
8 | +*.toc | |
9 | +*.fmt | |
10 | +*.fot | |
11 | +*.cb | |
12 | +*.cb2 | |
13 | +.*.lb | |
14 | + | |
15 | +## Intermediate documents: | |
16 | +*.dvi | |
17 | +*.xdv | |
18 | +*-converted-to.* | |
19 | +# these rules might exclude image files for figures etc. | |
20 | +# *.ps | |
21 | +# *.eps | |
22 | ||
23 | + | |
24 | +## Generated if empty string is given at "Please type another file name for output:" | |
25 | ||
26 | + | |
27 | +## Bibliography auxiliary files (bibtex/biblatex/biber): | |
28 | +*.bbl | |
29 | +*.bcf | |
30 | +*.blg | |
31 | +*-blx.aux | |
32 | +*-blx.bib | |
33 | +*.run.xml | |
34 | + | |
35 | +## Build tool auxiliary files: | |
36 | +*.fdb_latexmk | |
37 | +*.synctex | |
38 | +*.synctex(busy) | |
39 | +*.synctex.gz | |
40 | +*.synctex.gz(busy) | |
41 | +*.pdfsync | |
42 | + | |
43 | +## Build tool directories for auxiliary files | |
44 | +# latexrun | |
45 | +latex.out/ | |
46 | + | |
47 | +## Auxiliary and intermediate files from other packages: | |
48 | +# algorithms | |
49 | +*.alg | |
50 | +*.loa | |
51 | + | |
52 | +# achemso | |
53 | +acs-*.bib | |
54 | + | |
55 | +# amsthm | |
56 | +*.thm | |
57 | + | |
58 | +# beamer | |
59 | +*.nav | |
60 | +*.pre | |
61 | +*.snm | |
62 | +*.vrb | |
63 | + | |
64 | +# changes | |
65 | +*.soc | |
66 | + | |
67 | +# comment | |
68 | +*.cut | |
69 | + | |
70 | +# cprotect | |
71 | +*.cpt | |
72 | + | |
73 | +# elsarticle (documentclass of Elsevier journals) | |
74 | +*.spl | |
75 | + | |
76 | +# endnotes | |
77 | +*.ent | |
78 | + | |
79 | +# fixme | |
80 | +*.lox | |
81 | + | |
82 | +# feynmf/feynmp | |
83 | +*.mf | |
84 | +*.mp | |
85 | +*.t[1-9] | |
86 | +*.t[1-9][0-9] | |
87 | +*.tfm | |
88 | + | |
89 | +#(r)(e)ledmac/(r)(e)ledpar | |
90 | +*.end | |
91 | +*.?end | |
92 | +*.[1-9] | |
93 | +*.[1-9][0-9] | |
94 | +*.[1-9][0-9][0-9] | |
95 | +*.[1-9]R | |
96 | +*.[1-9][0-9]R | |
97 | +*.[1-9][0-9][0-9]R | |
98 | +*.eledsec[1-9] | |
99 | +*.eledsec[1-9]R | |
100 | +*.eledsec[1-9][0-9] | |
101 | +*.eledsec[1-9][0-9]R | |
102 | +*.eledsec[1-9][0-9][0-9] | |
103 | +*.eledsec[1-9][0-9][0-9]R | |
104 | + | |
105 | +# glossaries | |
106 | +*.acn | |
107 | +*.acr | |
108 | +*.glg | |
109 | +*.glo | |
110 | +*.gls | |
111 | +*.glsdefs | |
112 | +*.lzo | |
113 | +*.lzs | |
114 | +*.slg | |
115 | +*.slo | |
116 | +*.sls | |
117 | + | |
118 | +# uncomment this for glossaries-extra (will ignore makeindex's style files!) | |
119 | +# *.ist | |
120 | + | |
121 | +# gnuplot | |
122 | +*.gnuplot | |
123 | +*.table | |
124 | + | |
125 | +# gnuplottex | |
126 | +*-gnuplottex-* | |
127 | + | |
128 | +# gregoriotex | |
129 | +*.gaux | |
130 | +*.glog | |
131 | +*.gtex | |
132 | + | |
133 | +# htlatex | |
134 | +*.4ct | |
135 | +*.4tc | |
136 | +*.idv | |
137 | +*.lg | |
138 | +*.trc | |
139 | +*.xref | |
140 | + | |
141 | +# hyperref | |
142 | +*.brf | |
143 | + | |
144 | +# knitr | |
145 | +*-concordance.tex | |
146 | +# TODO Uncomment the next line if you use knitr and want to ignore its generated tikz files | |
147 | +# *.tikz | |
148 | +*-tikzDictionary | |
149 | + | |
150 | +# listings | |
151 | +*.lol | |
152 | + | |
153 | +# luatexja-ruby | |
154 | +*.ltjruby | |
155 | + | |
156 | +# makeidx | |
157 | +*.idx | |
158 | +*.ilg | |
159 | +*.ind | |
160 | + | |
161 | +# minitoc | |
162 | +*.maf | |
163 | +*.mlf | |
164 | +*.mlt | |
165 | +*.mtc[0-9]* | |
166 | +*.slf[0-9]* | |
167 | +*.slt[0-9]* | |
168 | +*.stc[0-9]* | |
169 | + | |
170 | +# minted | |
171 | +_minted* | |
172 | +*.pyg | |
173 | + | |
174 | +# morewrites | |
175 | +*.mw | |
176 | + | |
177 | +# newpax | |
178 | +*.newpax | |
179 | + | |
180 | +# nomencl | |
181 | +*.nlg | |
182 | +*.nlo | |
183 | +*.nls | |
184 | + | |
185 | +# pax | |
186 | +*.pax | |
187 | + | |
188 | +# pdfpcnotes | |
189 | +*.pdfpc | |
190 | + | |
191 | +# sagetex | |
192 | +*.sagetex.sage | |
193 | +*.sagetex.py | |
194 | +*.sagetex.scmd | |
195 | + | |
196 | +# scrwfile | |
197 | +*.wrt | |
198 | + | |
199 | +# svg | |
200 | +svg-inkscape/ | |
201 | + | |
202 | +# sympy | |
203 | +*.sout | |
204 | +*.sympy | |
205 | +sympy-plots-for-*.tex/ | |
206 | + | |
207 | +# pdfcomment | |
208 | +*.upa | |
209 | +*.upb | |
210 | + | |
211 | +# pythontex | |
212 | +*.pytxcode | |
213 | +pythontex-files-*/ | |
214 | + | |
215 | +# tcolorbox | |
216 | +*.listing | |
217 | + | |
218 | +# thmtools | |
219 | +*.loe | |
220 | + | |
221 | +# TikZ & PGF | |
222 | +*.dpth | |
223 | +*.md5 | |
224 | +*.auxlock | |
225 | + | |
226 | +# titletoc | |
227 | +*.ptc | |
228 | + | |
229 | +# todonotes | |
230 | +*.tdo | |
231 | + | |
232 | +# vhistory | |
233 | +*.hst | |
234 | +*.ver | |
235 | + | |
236 | +# easy-todo | |
237 | +*.lod | |
238 | + | |
239 | +# xcolor | |
240 | +*.xcp | |
241 | + | |
242 | +# xmpincl | |
243 | +*.xmpi | |
244 | + | |
245 | +# xindy | |
246 | +*.xdy | |
247 | + | |
248 | +# xypic precompiled matrices and outlines | |
249 | +*.xyc | |
250 | +*.xyd | |
251 | + | |
252 | +# endfloat | |
253 | +*.ttt | |
254 | +*.fff | |
255 | + | |
256 | +# Latexian | |
257 | +TSWLatexianTemp* | |
258 | + | |
259 | +## Editors: | |
260 | +# WinEdt | |
261 | +*.bak | |
262 | +*.sav | |
263 | + | |
264 | +# Texpad | |
265 | +.texpadtmp | |
266 | + | |
267 | +# LyX | |
268 | +*.lyx~ | |
269 | + | |
270 | +# Kile | |
271 | +*.backup | |
272 | + | |
273 | +# gummi | |
274 | +.*.swp | |
275 | + | |
276 | +# KBibTeX | |
277 | +*~[0-9]* | |
278 | + | |
279 | +# TeXnicCenter | |
280 | +*.tps | |
281 | + | |
282 | +# auto folder when using emacs and auctex | |
283 | +./auto/* | |
284 | +*.el | |
285 | + | |
286 | +# expex forward references with \gathertags | |
287 | +*-tags.tex | |
288 | + | |
289 | +# standalone packages | |
290 | +*.sta | |
291 | + | |
292 | +# Makeindex log files | |
293 | +*.lpz | |
294 | + | |
295 | +# xwatermark package | |
296 | +*.xwm | |
297 | + | |
298 | +# REVTeX puts footnotes in the bibliography by default, unless the nofootinbib | |
299 | +# option is specified. Footnotes are the stored in a file with suffix Notes.bib. | |
300 | +# Uncomment the next line to have this generated file ignored. | |
301 | +#*Notes.bib | ... | ... |
code/asp/bachelor.lp
code/asp/color.lp
code/asp/py_01.lp
... | ... | @@ -10,11 +10,13 @@ def gcd(a, b): |
10 | 10 | else: |
11 | 11 | na = a.number |
12 | 12 | nb = b.number |
13 | - nc = nb % na | |
14 | - return gcd(N(nc), a) | |
13 | + nc = N(nb % na) | |
14 | + return gcd(nc, a) | |
15 | 15 | |
16 | 16 | #end. |
17 | 17 | |
18 | 18 | p(210, 213). |
19 | 19 | p(1364, 385). |
20 | -gcd(X, Y, @gcd(X,Y)) :- p(X, Y). | |
21 | 20 | \ No newline at end of file |
21 | +gcd(X, Y, @gcd(X,Y)) :- p(X, Y). | |
22 | + | |
23 | +#show gcd/3. | |
22 | 24 | \ No newline at end of file | ... | ... |
code/asp/tutotial_01.lp
msc/Empty File
task_01/proposal.pdf
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1 | +\documentclass[a4paper]{article} | |
2 | + | |
3 | +\usepackage{commath} | |
4 | +\usepackage{hyperref} | |
5 | +\hypersetup{ | |
6 | + colorlinks=true, | |
7 | + linkcolor=blue, | |
8 | +} | |
9 | + | |
10 | +\title{Zugzwang\\\textit{Logic and Artificial Intelligence}} | |
11 | +\author{Francisco Coelho\\ \texttt{fc@uevora.pt}} | |
12 | + | |
13 | +\begin{document} | |
14 | +\maketitle | |
15 | + | |
16 | +\begin{abstract} | |
17 | + A major limitation of logical representations is the implicit assumption that the Background Knowledge (BK) is perfect. This assumption is problematic if data is noisy, which is often the case. Here we aim to explore how ASP specifications with probabilistic facts can lead to characterizations of probability functions on the specification's domain. | |
18 | +\end{abstract} | |
19 | + | |
20 | +\section{Introduction and Motivation } | |
21 | + | |
22 | +Answer Set Programming (ASP) is a logic programming paradigm based on the Stable Model semantics of Normal Logic Programs (NP) that can be implemented using the latest advances in SAT solving technology. ASP is a truly declarative language that supports language constructs such as disjunction in the head of a clause, choice rules, and hard and weak constraints. | |
23 | + | |
24 | +The Distribution Semantics (DS) is a key approach to extend logical representations with probabilistic reasoning. Probabilistic Facts (PF) are the most basic stochastic DS primitive and they take the form of logical facts, $a$, labelled with a probability, such as $p::a$; Each probabilistic fact represents a boolean random variable that is true with probability $p$ and false with probability $1 - p$. A (consistent) combination of the PFs defines a \textit{total choice} $\theta = \set{p::a, \ldots}$ such that | |
25 | + | |
26 | +\begin{equation} | |
27 | + P(\theta) = \prod_{a\in\theta} p \prod_{a \not\in \theta} (1- p). | |
28 | +\end{equation} | |
29 | + | |
30 | +Our goal is to extend this probability, from total choices, to cover the specification domain. We can foresee two key applications of this extended probability: | |
31 | + | |
32 | +\begin{enumerate} | |
33 | + \item Support any probabilistic reasoning/task on the specification domain. | |
34 | + \item Also, given a dataset and a divergence measure, now the specification can be scored (by the divergence w.r.t. the \emph{empiric} distribution of the dataset), and sorted amongst other specifications. This is a key ingredient in algorithms searching, for example, an \textit{optimal specification} of the dataset. | |
35 | +\end{enumerate} | |
36 | + | |
37 | +This goal faces a critical problem concerning situations where \textit{multiple} standard models result from a given total choice, illustrated by the following example. The specification | |
38 | +$$ | |
39 | +\begin{aligned} | |
40 | + 0.3::a&,\cr | |
41 | + b \vee c& \leftarrow a. | |
42 | +\end{aligned} | |
43 | +$$ | |
44 | +has three stable models, $\set{\neg a}, \set{a, b}$ and $\set{a, c}$. While it is straightforward to set $P(\neg a)=0.7$, there is \textit{no further information} to assign values to $P(a,b)$ and $P(a,c)$. At best, we can use a parameter $\alpha$ such that | |
45 | +$$ | |
46 | +\begin{aligned} | |
47 | +P(a,b) &= 0.3 \alpha,\cr | |
48 | +P(a,c) &= 0.3 (1 - \alpha). | |
49 | +\end{aligned} | |
50 | +$$ | |
51 | + | |
52 | +This uncertainty in inherent to the specification, but can be mitigated with the help of a dataset: the parameter $\alpha$ can be estimated from the empirical distribution. | |
53 | + | |
54 | +In summary, if an ASP specification is intended to describe some system that can be observed then: | |
55 | + | |
56 | +\begin{enumerate} | |
57 | + \item The observations can be used to estimate the value of the parameters (such as $\alpha$ above and others entailed from further clauses). | |
58 | + \item With a probability set for the stable models, we want to extend it to all the samples (\textit{i.e.} consistent sets of literals) of the specification. | |
59 | + \item This extended probability can then be related to the \textit{empirical distribution}, using a probability divergence, such as Kullback-Leibler; and the divergence value used as a \textit{performance} measure of the specification with respect to the observations. | |
60 | + \item If that specification is only but one of many possible candidates then that performance measure can be used, \textit{e.g.} as fitness, by algorithms searching (optimal) specifications of a dataset of observations. | |
61 | +\end{enumerate} | |
62 | + | |
63 | +Currently, we are on the step two above: Extending a probability function (with parameters such as $\alpha$), defined on the stable sets of a specification, to all the samples of the specification. This extension must, of course, respect the axioms of probability so that probabilistic reasoning is consistent with the ASP specification. | |
64 | + | |
65 | +\section{Work Plan} | |
66 | + | |
67 | +A team of two researchers and a undergraduate, master, or Ph.D. student, working over six months with adequate resources, should be able to advance substantial contributions and produce an intermediate progress report for a workshop, a final comprehensive paper for a conference, or start a Ph.D. project with greater reach and depth, describing: | |
68 | + | |
69 | +\begin{itemize} | |
70 | + \item The formalization of the methods outlined above, including the parameter estimation from observations and the probability distribution over the specification samples. | |
71 | + \item Application and evaluation of this approach, using tools such as \hyperlink{https://ciao-lang.org/playground/scasp.html}{s(casp)}, or the \hyperlink{https://potassco.org/}{Potassco suit} to a range of problems from the simple \textit{Burglar, Earthquake, Alarm} to measuring a specification accuracy on a given dataset, or finding an optimal specification for a given dataset given some background knowledge. | |
72 | +\end{itemize} | |
73 | + | |
74 | +While the theoretical word for this project has yet to be completed, there are some relevant tasks that, with different levels of ambition, can be solved right now: \marginpar{Estas tarefas precisam de maior\ldots\ folego.} | |
75 | +\begin{enumerate} | |
76 | + \item \textit{Extract Probability Annotations}. For example, convert the annotated specification \verb!0.3::a. b ; c :- a.! to \verb! a ; -a. b ; c :- a!. This is a simple, syntactical task that can be implemented either with \texttt{prolog} or using \texttt{python} and the API provided by the Potassco suite. | |
77 | + \item \textit{Extended Probability to Stable Models}. Application of the method outlined before, where the probability of total choices is extended to standard models using parameters, which are next estimated with a dataset. | |
78 | + \item \textit{Relate Samples, Stable Models and Total Choices}. Determine which stable models, or total choices, contain and which are contained in a given sample. | |
79 | +\end{enumerate} | |
80 | + | |
81 | +\section*{References} | |
82 | + | |
83 | + | |
84 | +\begin{enumerate} | |
85 | + \item Victor Verreet, Vincent Derkinderen, Pedro Zuidberg Dos Martires, Luc De Raedt, Inference and Learning with Model Uncertainty in Probabilistic Logic Programs (2022) | |
86 | + \item Andrew Cropper, Sebastijan Dumancic, Richard Evans, Stephen H. Muggleton, Inductive logic programming at 30 (2021) | |
87 | + \item Fabio Gagliardi Cozman, Denis Deratani Mauá, The joy of Probabilistic Answer Set Programming: Semantics - complexity, expressivity, inference (2020) | |
88 | + \item Fabrizio Riguzzi, Foundations of Probabilistic Logic Programming Languages, Semantics, Inference and Learning. Rivers Publishers (2018) | |
89 | + \item Martin Gebser, Roland Kaminski, Benjamin Kaufmann, and Torsten Schaub, Answer Set Solving in Practice, Morgan \& Claypool Publishers (2013) | |
90 | +\end{enumerate} | |
91 | + | |
92 | +\end{document} | |
0 | 93 | \ No newline at end of file | ... | ... |
text/00_PASP.aux
... | ... | @@ -1,170 +0,0 @@ |
1 | -\relax | |
2 | -\providecommand\hyper@newdestlabel[2]{} | |
3 | -\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument} | |
4 | -\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined | |
5 | -\global\let\oldcontentsline\contentsline | |
6 | -\gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}} | |
7 | -\global\let\oldnewlabel\newlabel | |
8 | -\gdef\newlabel#1#2{\newlabelxx{#1}#2} | |
9 | -\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}} | |
10 | -\AtEndDocument{\ifx\hyper@anchor\@undefined | |
11 | -\let\contentsline\oldcontentsline | |
12 | -\let\newlabel\oldnewlabel | |
13 | -\fi} | |
14 | -\fi} | |
15 | -\global\let\hyper@last\relax | |
16 | -\gdef\HyperFirstAtBeginDocument#1{#1} | |
17 | -\providecommand*\HyPL@Entry[1]{} | |
18 | -\HyPL@Entry{0<</P(\376\377\0001)>>} | |
19 | -\@writefile{nav}{\headcommand {\slideentry {0}{0}{1}{1/1}{}{0}}} | |
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text/00_PASP.nav
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23 | 23 | \usepackage[overridenote]{pdfpc} |
24 | 24 | |
25 | 25 | \usepackage{tikz} |
26 | +\usetikzlibrary{ | |
27 | + % shapes, | |
28 | + % arrows, | |
29 | + % backgrounds, | |
30 | + positioning, | |
31 | +} | |
26 | 32 | \tikzset{ |
27 | -rrect/.style={ | |
28 | - rectangle, | |
29 | - rounded corners, | |
30 | - fill=gray!10, | |
31 | - % minimum width=1em, | |
32 | - % text width=2em, | |
33 | - % draw=blue!75, | |
34 | - % very thick, | |
35 | - align=center, | |
36 | - } | |
33 | + sm/.style={ | |
34 | + rectangle, | |
35 | + rounded corners, | |
36 | + fill=teal!10, | |
37 | + align=center, | |
38 | + minimum height=1.5em, | |
39 | + }, | |
40 | + sup/.style={ | |
41 | + rectangle, | |
42 | + rounded corners, | |
43 | + fill=blue!10, | |
44 | + align=center, | |
45 | + minimum height=1.5em, | |
46 | + }, | |
47 | + sub/.style={ | |
48 | + rectangle, | |
49 | + rounded corners, | |
50 | + fill=red!10, | |
51 | + align=center, | |
52 | + minimum height=1.5em, | |
53 | + }, | |
54 | + ind/.style={ | |
55 | + align=center, | |
56 | + minimum height=1.5em, | |
57 | + }, | |
37 | 58 | } |
38 | 59 | |
39 | -\usepackage{commath} | |
40 | -\usepackage{amssymb} | |
41 | 60 | \usepackage[T1]{fontenc} |
42 | 61 | \usepackage{hyperref} |
43 | 62 | \hypersetup{% |
44 | 63 | colorlinks=true, |
45 | 64 | allcolors=blue, |
46 | 65 | } |
66 | +\usepackage{commath} | |
67 | +\usepackage{amssymb} | |
47 | 68 | % |
48 | 69 | % Local commands |
49 | 70 | % |
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89 | 110 | % -------------------------------- |
90 | 111 | \item $\co{x} = 1 - x$. |
91 | 112 | % -------------------------------- |
92 | - \item \textbf{Probabilistic Atomic Choice (PAC):} $\alpha :: a$ defines $a \lor \neg a$ and probabilities $\pr{a} = \alpha, \pr{\neg a} = \co{\alpha}$. | |
113 | + \item \textbf{Probabilistic Atomic Choice (PAC):} $x :: a$ defines $a \lor \neg a$ and probabilities $\pr{a} = x, \pr{\neg a} = \co{x}$. | |
93 | 114 | % -------------------------------- |
94 | - \item $\delta a$ denotes $a \lor \neg a$ and $\delta\! \set{\alpha :: a, a \in \fml{A}} = \set{\delta a, a \in \fml{A}}$ for a set of atoms $\fml{A}$. | |
115 | + \item $\delta a$ denotes $a \lor \neg a$ and $\delta\! \set{x :: a, a \in \fml{A}} = \set{\delta a, a \in \fml{A}}$ for a set of atoms $\fml{A}$. | |
95 | 116 | % -------------------------------- |
96 | - \item \textbf{Closed World Assumption:} $\naf x \models \neg x$. | |
117 | + \item \textbf{Closed World Assumption:} $\naf p \models \neg p$. | |
97 | 118 | % -------------------------------- |
98 | 119 | % \item Probabilistic choices and sub-goals are independent. |
99 | 120 | % -------------------------------- |
... | ... | @@ -106,25 +127,24 @@ rrect/.style={ |
106 | 127 | \begin{itemize} |
107 | 128 | % -------------------------------- |
108 | 129 | \item \textbf{Atoms} $\fml{A}$, |
109 | - $\overline{\fml{A}} = \cset{\neg a}{a \in \fml{A}}$, | |
130 | + $\overline{\fml{A}} = \cset{\neg a}{a \in \fml{A}}$, and \textbf{literals} $\fml{L} = \fml{A} \cup \co{\fml{A}}$. | |
110 | 131 | % -------------------------------- |
111 | - \item \textbf{Samples} $\fml{Z}$: | |
112 | - $$\fml{Z} = \cset{z = \alpha \cup \beta }{ \alpha \subseteq \fml{A} \land \beta \subseteq \overline{\fml{A}} }$$ | |
132 | + \item \textbf{Samples} $z \in \fml{Z} \iff z \subseteq \fml{L}$. | |
113 | 133 | % -------------------------------- |
114 | - \item \textbf{Interpretations} or \textit{consistent samples} $\fml{I}$ : | |
115 | - $$\fml{I} = \cset{z \in \fml{Z} }{ \forall a \in \fml{A}~\envert{\set{a,\neg a} \cap z} \leq 1}.$$ | |
134 | + \item \textbf{Events} or \textit{consistent samples} $\fml{E}$ : | |
135 | + $$\fml{E} = \cset{z \in \fml{Z} }{ \forall a \in \fml{A}~\envert{\set{a,\neg a} \cap z} \leq 1}.$$ | |
116 | 136 | % -------------------------------- |
117 | 137 | \item \textit{PASP Problem} or \textbf{Specification:} $P = C \land F \land R$ where |
118 | 138 | % -------------------------------- |
119 | 139 | \begin{itemize} |
120 | 140 | % -------------------------------- |
121 | - \item $C = C_P = \cset{\alpha_i :: a_i }{ i \in 1:n \land a_i \in \fml{A}}$ \textit{pacs}. | |
141 | + \item $C = C_P = \cset{x_i :: a_i }{ i \in 1:n \land a_i \in \fml{A}}$ \textit{pacs}. | |
122 | 142 | % -------------------------------- |
123 | 143 | \item $F = F_P$ \textit{facts}. |
124 | 144 | % -------------------------------- |
125 | 145 | \item $R = R_P$ \textit{rules}. |
126 | 146 | % -------------------------------- |
127 | - \item $\fml{A}_P, \fml{Z}_P$ and $\fml{I}_P$: \textit{atoms}, \textit{samples} and \textit{interpretations} of $P$. | |
147 | + \item $\fml{A}_P, \fml{Z}_P$ and $\fml{E}_P$: \textit{atoms}, \textit{samples} and \textit{events} of $P$. | |
128 | 148 | \end{itemize} |
129 | 149 | % -------------------------------- |
130 | 150 | \item \textbf{Stable Models} of $P$, $\fml{S} = \fml{S}_P$, are the stable models of $\delta P = \delta C + F + R$. |
... | ... | @@ -137,7 +157,7 @@ rrect/.style={ |
137 | 157 | % -------------------------------- |
138 | 158 | \begin{itemize} |
139 | 159 | % -------------------------------- |
140 | - \item \textbf{Total Choices:} $\Theta = \Theta_C = \Theta_P$ elements are $\theta = \cset{t_c}{c \in C}$ where $c=\alpha::a$ and $t_c$ is $a$ or $\neg a$. | |
160 | + \item \textbf{Total Choices:} $\Theta = \Theta_C = \Theta_P$ elements are $\theta = \cset{t_c}{c \in C}$ where $c=x::a$ and $t_c$ is $a$ or $\neg a$. | |
141 | 161 | % -------------------------------- |
142 | 162 | %\item For $s\in\fml{S}$ let $\theta_s \subseteq s$ (unique \textit{total choice}) |
143 | 163 | %\item Define $\fml{S}_\theta = \cset{s \in \fml{S}}{\theta \subset s}$. |
... | ... | @@ -146,7 +166,7 @@ rrect/.style={ |
146 | 166 | % -------------------------------- |
147 | 167 | \item \textbf{Total Choice Probability:} |
148 | 168 | \begin{equation} |
149 | - \pr{\theta} = \prod_{a \in \theta}\alpha \prod_{\neg a \in \theta}\co{\alpha}.\label{eq:prob.tc} | |
169 | + \pr{\theta} = \prod_{a \in \theta}x \prod_{\neg a \in \theta}\co{x}.\label{eq:prob.tc} | |
150 | 170 | \end{equation} |
151 | 171 | % -------------------------------- |
152 | 172 | \end{itemize} |
... | ... | @@ -157,7 +177,7 @@ rrect/.style={ |
157 | 177 | \begin{frame} |
158 | 178 | % -------------------------------- |
159 | 179 | \begin{block}{Problem Statement} |
160 | - How to \textit{extend} probability from total choices to stable models, interpretations and samples? | |
180 | + How to \textit{extend} probability from total choices to stable models, events and samples? | |
161 | 181 | \end{block} |
162 | 182 | % -------------------------------- |
163 | 183 | \begin{quotation} |
... | ... | @@ -193,11 +213,11 @@ rrect/.style={ |
193 | 213 | % -------------------------------- |
194 | 214 | \item $\pr{\set{\neg a}} = 0.7$ is straightforward. |
195 | 215 | % -------------------------------- |
196 | - \item But, no \textit{informed} choice for $\alpha\in\intcc{0,1}$ in | |
216 | + \item But, no \textit{informed} choice for $x\in\intcc{0,1}$ in | |
197 | 217 | $$ |
198 | 218 | \begin{aligned} |
199 | - \pr{\set{a, b}} &= 0.3 \alpha, \cr | |
200 | - \pr{\set{a, c}} &= 0.3 \co{\alpha}. | |
219 | + \pr{\set{a, b}} &= 0.3 x, \cr | |
220 | + \pr{\set{a, c}} &= 0.3 \co{x}. | |
201 | 221 | \end{aligned} |
202 | 222 | $$ |
203 | 223 | % -------------------------------- |
... | ... | @@ -209,28 +229,43 @@ rrect/.style={ |
209 | 229 | % -------------------------------- |
210 | 230 | \begin{itemize} |
211 | 231 | % -------------------------------- |
212 | - \item The specification \textit{lacks information} to set $\alpha\in\intcc{0,1}$ in | |
232 | + \item The specification \textit{lacks information} to set $x\in\intcc{0,1}$ in | |
213 | 233 | $$ |
214 | 234 | \begin{aligned} |
215 | - \pr{\set{a, b}} &= 0.3 \alpha, \cr | |
216 | - \pr{\set{a, c}} &= 0.3 \co{\alpha}. | |
235 | + \pr{\set{a, b}} &= 0.3 x, \cr | |
236 | + \pr{\set{a, c}} &= 0.3 \co{x}. | |
217 | 237 | \end{aligned} |
218 | 238 | $$ |
219 | - \item A \textit{random variable} captures this uncertainty: | |
239 | + \item A \textit{random variable} captures this uncertainty, \alert{assuming} that the stable models are statistically independent: | |
220 | 240 | $$ |
221 | 241 | \begin{aligned} |
222 | - \pr{\set{\neg a} \given A = \alpha } &= 0.7, \cr | |
223 | - \pr{\set{a, b} \given A = \alpha } &= 0.3 \alpha, \cr | |
224 | - \pr{\set{a, c} \given A = \alpha } &= 0.3 \co{\alpha}. | |
242 | + \pr{\set{\neg a} \given X = x } &= 0.7, \cr | |
243 | + \pr{\set{a, b} \given X = x } &= 0.3 x, \cr | |
244 | + \pr{\set{a, c} \given X = x } &= 0.3 \co{x}. | |
225 | 245 | \end{aligned} |
226 | 246 | $$ |
227 | - \item Other uncertainties lead to further parameters: | |
247 | + \item Other uncertainties may lead to further conditions: | |
228 | 248 | $$ |
229 | - \pr{s \given A_1 = \alpha_1, \ldots, A_n = \alpha_n }. | |
249 | + \pr{s \given X_1 = x_1, \ldots, X_n = x_n }. | |
230 | 250 | $$ |
231 | 251 | % -------------------------------- |
232 | 252 | \end{itemize} |
233 | - Reducing \textbf{specification uncertainty}, \textit{e.g.} setting $A = 0.21$, must result from \textbf{observations}. | |
253 | + Reducing \textbf{uncertainty}, \textit{e.g.} setting $X = 0.21$, must result from \textbf{external} sources, since the specification lacks information for further assertions. | |
254 | + % -------------------------------- | |
255 | +\end{frame} | |
256 | +% ================================================================ | |
257 | +\begin{frame}{Independence of Stable Models} | |
258 | + % -------------------------------- | |
259 | + | |
260 | + \begin{itemize} | |
261 | + \item[Q:] Why are the stable models assumed statistically independent? | |
262 | + % -------------------------------- | |
263 | + \item[A:] Because dependence can be \textit{explicitly} modelled. | |
264 | + % -------------------------------- | |
265 | + \item So, it is assumed \textit{intention} of the \textit{modeller} to not explicit express such dependences. | |
266 | + % -------------------------------- | |
267 | + \item \textbf{For example:} \todo{Some key examples}. | |
268 | + \end{itemize} | |
234 | 269 | % -------------------------------- |
235 | 270 | \end{frame} |
236 | 271 | % ================================================================ |
... | ... | @@ -239,9 +274,9 @@ rrect/.style={ |
239 | 274 | A \textit{random variable} captures this uncertainty: |
240 | 275 | $$ |
241 | 276 | \begin{aligned} |
242 | - \pr{\set{\neg a} \given A = \alpha } &= 0.7, \cr | |
243 | - \pr{\set{a, b} \given A = \alpha } &= 0.3 \alpha, \cr | |
244 | - \pr{\set{a, c} \given A = \alpha } &= 0.3 \co{\alpha}. | |
277 | + \pr{\set{\neg a} \given X = x } &= 0.7, \cr | |
278 | + \pr{\set{a, b} \given X = x } &= 0.3 x, \cr | |
279 | + \pr{\set{a, c} \given X = x } &= 0.3 \co{x}. | |
245 | 280 | \end{aligned} |
246 | 281 | $$ |
247 | 282 | % -------------------------------- |
... | ... | @@ -265,13 +300,13 @@ rrect/.style={ |
265 | 300 | % -------------------------------- |
266 | 301 | \end{frame} |
267 | 302 | % ================================================================ |
268 | -\begin{frame}%{Specification, Data \& Evaluation} | |
303 | +\begin{frame}{Leap into Inductive Programming} | |
269 | 304 | % -------------------------------- |
270 | 305 | Given a method that produces a distribution of samples, $p$, from a specification, $P$ and: |
271 | 306 | % -------------------------------- |
272 | 307 | \begin{itemize} |
273 | 308 | % -------------------------------- |
274 | - \item $Z$, a dataset of samples. | |
309 | + \item $Z$, a dataset (of samples). | |
275 | 310 | % -------------------------------- |
276 | 311 | \item $e$, the respective empirical distribution. |
277 | 312 | % -------------------------------- |
... | ... | @@ -279,8 +314,18 @@ rrect/.style={ |
279 | 314 | % -------------------------------- |
280 | 315 | \end{itemize} |
281 | 316 | % -------------------------------- |
282 | - \begin{block}{Motivation} | |
283 | - Then $D\at{P} = D\at{e, p}$ is a \textbf{performance} measure of $P$ and can be used, \textit{e.g.} as fitness, by algorithms searching for \textbf{optimal specifications of a dataset}. | |
317 | + \begin{block}{Specification Performance \& Inductive Programming} | |
318 | + % -------------------------------- | |
319 | + \begin{itemize} | |
320 | + % -------------------------------- | |
321 | + \item $D\at{P} = D\at{e, p}$ is a \textbf{performance} measure of $P$. | |
322 | + % -------------------------------- | |
323 | + \item Predictor performance measures, such as accuracy, are common in \textit{Machine Learning} tasks. | |
324 | + % -------------------------------- | |
325 | + \item For \textit{Inductive Programming} this performance can be used, \textit{e.g.} as fitness, by algorithms searching for \textbf{optimal specifications of a dataset}. | |
326 | + % -------------------------------- | |
327 | + \end{itemize} | |
328 | + % -------------------------------- | |
284 | 329 | \end{block} |
285 | 330 | % -------------------------------- |
286 | 331 | \end{frame} |
... | ... | @@ -290,18 +335,21 @@ rrect/.style={ |
290 | 335 | \begin{frame}{Resolution Path} |
291 | 336 | Prior to \textit{conciliation} with data: |
292 | 337 | \begin{enumerate} |
293 | - \item Use \textit{conditional parameters} to represent lack of knowledge \emph{e.g.} $\alpha$ in the disjunction example. | |
294 | - \item This extends probability from total choices to \textit{standard models} --- \alert{hopefully}. | |
295 | - \item Assume probability set on standard models; Extend it to \textit{interpretations}. \textbf{How?} Later. | |
296 | - \item Set $\pr{z} = 0$ for $z \in \fml{Z} \setminus \fml{I}$ (inconsistent samples). | |
338 | + \item \alert{Hopefully}, \textit{conditional parameters} extend probability from total choices to \textit{standard models}. | |
339 | + \item \textbf{How} to extend it to \textit{events}? | |
340 | + \begin{itemize} | |
341 | + \item $\pr{x} = 0$ for $x$ \textit{excluded} by the specification, including \textit{inconsistent} samples. | |
342 | + \item $\pr{x}$ depends on the $s \in \fml{S}$ that contain/are contained in $x$. | |
343 | + \end{itemize} | |
297 | 344 | \end{enumerate} |
298 | - Assuming a conditional probability set on stable models, \textit{How to extend it to interpretations?} | |
345 | + \alert{Consider probabilities \textbf{conditional} on the total choice!} | |
299 | 346 | \end{frame} |
300 | -\begin{frame}{Bounds of Interpretations} | |
347 | +% ================================================================ | |
348 | +\begin{frame}{Bounds of Events} | |
301 | 349 | % -------------------------------- |
302 | 350 | \begin{itemize} |
303 | 351 | % -------------------------------- |
304 | - \item For $x\in\fml{I}$: | |
352 | + \item For $x\in\fml{E}$: | |
305 | 353 | % -------------------------------- |
306 | 354 | \begin{itemize} |
307 | 355 | % -------------------------------- |
... | ... | @@ -315,13 +363,25 @@ rrect/.style={ |
315 | 363 | % -------------------------------- |
316 | 364 | \begin{enumerate} |
317 | 365 | % -------------------------------- |
318 | - \item\label{prop:lucases.a} $\lset{x} = \set{x} = \uset{x}$ and $x$ is a stable model. | |
366 | + \item\label{prop:lucases.a} $\lset{x} = \set{x} = \uset{x}$ and $x$ is a stable model. Then: | |
367 | + \begin{equation} | |
368 | + \pr{x \given C = \theta_x} = d\at{x}. | |
369 | + \end{equation} | |
319 | 370 | % -------------------------------- |
320 | - \item\label{prop:lucases.b} $\lset{x} \neq \emptyset \land \uset{x} = \emptyset$. | |
371 | + \item\label{prop:lucases.b} $\lset{x} \neq \emptyset \land \uset{x} = \emptyset$. Then: | |
372 | + \begin{equation} | |
373 | + \pr{x \given C = \theta_s, s \in \lset{x}} = \prod_{s\in\lset{x}} d\at{s}. | |
374 | + \end{equation} | |
321 | 375 | % -------------------------------- |
322 | - \item\label{prop:lucases.c} $\lset{x} = \emptyset \land \uset{x} \neq \emptyset$. | |
376 | + \item\label{prop:lucases.c} $\lset{x} = \emptyset \land \uset{x} \neq \emptyset$. Then: | |
377 | + \begin{equation} | |
378 | + \pr{x \given C = \theta_s, s \in \uset{x}} = \sum_{s\in\uset{x}} d\at{s}. | |
379 | + \end{equation} | |
323 | 380 | % -------------------------------- |
324 | - \item\label{prop:lucases.d} $\lset{x} = \emptyset = \uset{x}$. | |
381 | + \item\label{prop:lucases.d} $\lset{x} = \emptyset = \uset{x}$. Then: | |
382 | + \begin{equation} | |
383 | + \pr{x} = 0. | |
384 | + \end{equation} | |
325 | 385 | % -------------------------------- |
326 | 386 | \end{enumerate} |
327 | 387 | because stable models are \textit{minimal}. |
... | ... | @@ -330,57 +390,326 @@ rrect/.style={ |
330 | 390 | % -------------------------------- |
331 | 391 | \end{frame} |
332 | 392 | % ================================================================ |
333 | -\begin{frame}{Disjunction Example | The Interpretation's Lattice} | |
393 | +\begin{frame}{Conditional on Total Choices} | |
394 | + % -------------------------------- | |
395 | + \begin{itemize} | |
396 | + % -------------------------------- | |
397 | + \item A stable model is entailed by an atomic choice plus the facts and rules of the specification. | |
398 | + \item We express that entailment as a \textit{conditional}. For example: | |
399 | + $$\pr{\set{a,b} \given X = x} = \pr{b \given X = x, \Theta = a}\pr{\theta = a}$$ | |
400 | + \item And now $\pr{b \given X = x, \Theta = a} = x$, since $X$ is a proxy for the stable models of the total choice $\theta = a$, we can further. | |
401 | + % -------------------------------- | |
402 | + \end{itemize} | |
403 | + % -------------------------------- | |
404 | +\end{frame} | |
405 | +% ================================================================ | |
406 | +\begin{frame}{Disjunction Example | The Events Lattice} | |
334 | 407 | \begin{center} |
335 | 408 | \begin{tikzpicture} |
409 | + % -------------------------------- | |
410 | + %\draw [help lines, color=gray!20] grid (11,7); | |
411 | + % -------------------------------- | |
412 | + \node at (7, 7) {$\pr{\Theta=a} = 0.3$}; | |
413 | + \node at (7, 6) {$x = \pr{S = ab \given \Theta}$}; | |
414 | + \node at (7, 5) {$\co{x} = \pr{S \not= ab \given \Theta}$}; | |
415 | + \node at (7, 7.5) {$\pr{E = abc \given \Theta} = \pr{S = ab, S = ac \given \Theta }$}; | |
416 | + % -------------------------------- | |
417 | + % \node [rrect] (sub) at (2, 7) {sub}; | |
418 | + % -------------------------------- | |
419 | + % \node [ fill=gray!10] (sup) at (3, 7) {sup}; | |
420 | + % -------------------------------- | |
421 | + % \node (ind) at (4, 7) {ind}; | |
422 | + % -------------------------------- | |
336 | 423 | % |
337 | - \draw [help lines, color=gray!20] grid (11,7); | |
338 | - % | |
339 | - \node [fill=gray!50] (sm) at (1, 7) {sm}; | |
340 | - \node [draw,rrect] (sub) at (2, 7) {sub}; | |
341 | - \node [draw, fill=gray!10] (sup) at (3, 7) {sup}; | |
342 | - \node (ind) at (4, 7) {ind}; | |
343 | - % | |
344 | - \node [draw, rrect] (E) at (5.5,0) {$\emptyset$}; | |
424 | + % -------------------------------- | |
425 | + \node [ sub, | |
426 | + pin=45:\textcolor{violet}{$1$} ] | |
427 | + (E) at (5.5,0) {$\emptyset$}; | |
428 | + % -------------------------------- | |
345 | 429 | % |
346 | - \node [draw, rrect] (a) at (1.5,1.5) {$a$}; | |
347 | - \node [draw, rrect] (b) at (0,1.5) {$b$}; | |
348 | - \node [draw, rrect] (c) at (4.5,1.5) {$c$}; | |
349 | - \node [fill=gray!50] (A) at (8.5,1.5) {$\co{a}$}; | |
350 | - \node (B) at (9.5,1.5) {$\co{b}$}; | |
351 | - \node (C) at (10.5,1.5) {$\co{c}$}; | |
430 | + % -------------------------------- | |
431 | + \node [ sub, | |
432 | + pin=0:\textcolor{blue!75}{$1$} ] | |
433 | + (a) at (1.5,1.5) {$a$}; | |
434 | + % -------------------------------- | |
435 | + \node [ sub, | |
436 | + pin=315:\textcolor{blue!50}{$x$}] | |
437 | + (b) at (0,1.5) {$b$}; | |
438 | + % -------------------------------- | |
439 | + \node [ sub, | |
440 | + pin=315:\textcolor{blue!50}{$\co{x}$}] | |
441 | + (c) at (4.5,1.5) {$c$}; | |
442 | + % -------------------------------- | |
443 | + % \node [ sm, | |
444 | + % pin=270:\textcolor{teal}{$1$}] | |
445 | + % (A) at (8.5,1.5) {$\co{a}$}; | |
446 | + % % -------------------------------- | |
447 | + % \node [ ind, | |
448 | + % pin=270:\textcolor{purple}{$0$}] | |
449 | + % (B) at (9.5, 1.5) {$\co{b}$}; | |
450 | + % % -------------------------------- | |
451 | + % \node [ ind, | |
452 | + % pin=270:\textcolor{purple}{$0$}] | |
453 | + % (C) at (10.5, 1.5) {$\co{c}$}; | |
454 | + % -------------------------------- | |
352 | 455 | % |
353 | - \node [fill=gray!50] (ab) at (0,4) {$ab$}; | |
354 | - \node [fill=gray!50] (ac) at (3,4) {$ac$}; | |
355 | - \node (aB) at (1,4) {$a\co{b}$}; | |
356 | - \node (aC) at (2,4) {$a\co{c}$}; | |
357 | - \node [draw, fill=gray!10] (Ab) at (4,4) {$\co{a}b$}; | |
358 | - \node [draw, fill=gray!10] (Ac) at (5,4) {$\co{a}c$}; | |
359 | - \node [draw, fill=gray!10] (AB) at (6,4) {$\co{a}\co{b}$}; | |
360 | - \node [draw, fill=gray!10] (AC) at (7,4) {$\co{a}\co{c}$}; | |
361 | - \node [fill=white] (bc) at (10,4) {$bc$}; | |
362 | - \node [fill=white] (bC) at (11,4) {$b\co{c}$}; | |
363 | - \node [fill=white] (Bc) at (12,4) {$\co{b}c$}; | |
364 | - \node [fill=white] (BC) at (13,4) {$\co{b}\co{c}$}; | |
456 | + % -------------------------------- | |
457 | + \node [ sm, | |
458 | + pin=0:\textcolor{teal}{$x$}] | |
459 | + (ab) at (0,4) {$ab$}; | |
460 | + % -------------------------------- | |
461 | + \node [ sm, | |
462 | + pin=0:\textcolor{teal}{$\co{x}$}] | |
463 | + (ac) at (3,4) {$ac$}; | |
464 | + % -------------------------------- | |
465 | + % \node [ ind, | |
466 | + % pin=270:\textcolor{purple}{$0$}] | |
467 | + % (aB) at (1,4) {$a\co{b}$}; | |
468 | + % % -------------------------------- | |
469 | + % \node [ ind, | |
470 | + % pin=270:\textcolor{purple}{$0$}] | |
471 | + % (aC) at (2,4) {$a\co{c}$}; | |
472 | + % -------------------------------- | |
473 | + % \node [ sup, | |
474 | + % pin=90:\textcolor{blue!50}{$1$}] | |
475 | + % (Ab) at (4,4) {$\co{a}b$}; | |
476 | + % % -------------------------------- | |
477 | + % \node [ sup, | |
478 | + % pin=90:\textcolor{blue!50}{$1$}] | |
479 | + % (Ac) at (5,4) {$\co{a}c$}; | |
480 | + % % -------------------------------- | |
481 | + % \node [ sup, | |
482 | + % pin=90:\textcolor{blue!50}{$1$}] | |
483 | + % (AB) at (6,4) {$\co{a}\co{b}$}; | |
484 | + % % -------------------------------- | |
485 | + % \node [ sup, | |
486 | + % pin=90:\textcolor{blue!50}{$1$}] | |
487 | + % (AC) at (7,4) {$\co{a}\co{c}$}; | |
488 | + % % -------------------------------- | |
489 | + % \node [ ind, | |
490 | + % pin=270:\textcolor{purple}{$0$}] | |
491 | + % (bc) at (10,4) {$bc$}; | |
492 | + % % -------------------------------- | |
493 | + % \node [ ind, | |
494 | + % pin=270:\textcolor{purple}{$0$}] | |
495 | + % (bC) at (11,4) {$b\co{c}$}; | |
496 | + % % -------------------------------- | |
497 | + % \node [ ind, | |
498 | + % pin=270:\textcolor{purple}{$0$}] | |
499 | + % (Bc) at (9.5,3.5) {$\co{b}c$}; | |
500 | + % % -------------------------------- | |
501 | + % \node [ ind, | |
502 | + % pin=270:\textcolor{purple}{$0$}] | |
503 | + % (BC) at (10.5,3.5) {$\co{b}\co{c}$}; | |
504 | + % -------------------------------- | |
365 | 505 | % |
366 | - \node [draw, fill=gray!10] (abc) at (1.5,6) {$abc$}; | |
367 | - \node (abC) at (3,6) {$ab\co{c}$}; | |
368 | - \node (aBc) at (4,6) {$a\co{b}c$}; | |
369 | - \node (aBC) at (5,6) {$a\co{b}\co{c}$}; | |
370 | - \node [draw, fill=gray!10] (Abc) at (7,6) {$\co{a}bc$}; | |
371 | - \node [draw, fill=gray!10] (AbC) at (8,6) {$\co{a}b\co{c}$}; | |
372 | - \node [draw, fill=gray!10] (ABc) at (9,6) {$\co{a}\co{b}c$}; | |
373 | - \node [draw, fill=gray!10] (ABC) at (10,6) {$\co{a}\co{b}\co{c}$}; | |
506 | + % -------------------------------- | |
507 | + \node [ sup, | |
508 | + pin=45:\textcolor{blue!75}{$x\co{x}$}] | |
509 | + (abc) at (1.5,6) | |
510 | + {$abc$}; | |
511 | + % -------------------------------- | |
512 | + \node [ sup, | |
513 | + pin=45:\textcolor{blue!50}{$x$}] | |
514 | + (abC) at (0,6) {$ab\co{c}$}; | |
515 | + % -------------------------------- | |
516 | + \node [ sup, | |
517 | + pin=45:\textcolor{blue!50}{$\co{x}$}] | |
518 | + (aBc) at (3,6) {$a\co{b}c$}; | |
519 | + % -------------------------------- | |
520 | + % \node [ ind, | |
521 | + % pin=90:\textcolor{purple}{$0$}] | |
522 | + % (aBC) at (5,6) {$a\co{b}\co{c}$}; | |
523 | + % -------------------------------- | |
524 | + % \node [ sup, | |
525 | + % pin=90:\textcolor{blue!50}{$1$}] | |
526 | + % (Abc) at (7,6) {$\co{a}bc$}; | |
527 | + % % -------------------------------- | |
528 | + % \node [ sup, | |
529 | + % pin=270:\textcolor{blue!50}{$1$}] | |
530 | + % (AbC) at (8,6) {$\co{a}b\co{c}$}; | |
531 | + % % -------------------------------- | |
532 | + % \node [ sup, | |
533 | + % pin=270:\textcolor{blue!50}{$1$}] | |
534 | + % (ABc) at (9,6) {$\co{a}\co{b}c$}; | |
535 | + % % -------------------------------- | |
536 | + % \node [ sup, | |
537 | + % pin=270:\textcolor{blue!50}{$1$}] | |
538 | + % (ABC) at (10,6) {$\co{a}\co{b}\co{c}$}; | |
539 | + % -------------------------------- | |
374 | 540 | % |
541 | + % -------------------------------- | |
375 | 542 | \draw [->] (ab) to [out=270,in=180] (E); |
376 | 543 | \draw [->] (ab) to [out=270,in=90] (a); |
377 | 544 | \draw [->] (ab) to [out=270,in=90] (b); |
378 | 545 | \draw [->] (ab) to [out=90,in=270] (abc); |
546 | + \draw [->] (ab) to [out=90,in=270] (abC); | |
379 | 547 | % |
380 | 548 | \draw [->] (ac) to [out=270,in=180] (E); |
381 | 549 | \draw [->] (ac) to [out=270,in=90] (a); |
382 | 550 | \draw [->] (ac) to [out=270,in=90] (c); |
383 | 551 | \draw [->] (ac) to [out=90,in=270] (abc); |
552 | + \draw [->] (ac) to [out=90,in=270] (aBc); | |
553 | + % | |
554 | + % \draw [->] (A) to [out=270,in=0] (E); | |
555 | + % % | |
556 | + % \draw [->] (A) to [out=90,in=270] (Abc); | |
557 | + % \draw [->] (A) to [out=90,in=270] (AbC); | |
558 | + % \draw [->] (A) to [out=90,in=270] (ABc); | |
559 | + % \draw [->] (A) to [out=90,in=270] (ABC); | |
560 | + % % | |
561 | + % \draw [->] (A) to [out=90,in=270] (Ab); | |
562 | + % \draw [->] (A) to [out=90,in=270] (Ac); | |
563 | + % \draw [->] (A) to [out=90,in=270] (AB); | |
564 | + % \draw [->] (A) to [out=90,in=270] (AC); | |
565 | + \end{tikzpicture} | |
566 | + \end{center} | |
567 | +\end{frame} | |
568 | +% ================================================================ | |
569 | +\begin{frame}{Disjunction Example | The Events Lattice} | |
570 | + \begin{center} | |
571 | + \begin{tikzpicture} | |
572 | + % -------------------------------- | |
573 | + %\draw [help lines, color=gray!20] grid (11,7); | |
574 | + % -------------------------------- | |
575 | + \node [sm] (sm) at (5, 7) {$\pr{\Theta=\set{\co{a}}} = \co{0.3}$}; | |
576 | + % -------------------------------- | |
577 | + % \node [rrect] (sub) at (2, 7) {sub}; | |
578 | + % -------------------------------- | |
579 | + % \node [ fill=gray!10] (sup) at (3, 7) {sup}; | |
580 | + % -------------------------------- | |
581 | + % \node (ind) at (4, 7) {ind}; | |
582 | + % -------------------------------- | |
583 | + % | |
584 | + % -------------------------------- | |
585 | + \node [ sub, | |
586 | + pin=45:\textcolor{violet}{$1$} ] | |
587 | + (E) at (5.5,0) {$\emptyset$}; | |
588 | + % -------------------------------- | |
589 | + % | |
590 | + % -------------------------------- | |
591 | + % \node [ sub, | |
592 | + % pin=270:\textcolor{blue!75}{$1$} ] | |
593 | + % (a) at (1.5,1.5) {$a$}; | |
594 | + % % -------------------------------- | |
595 | + % \node [ sub, | |
596 | + % pin=270:\textcolor{blue!50}{$x$}] | |
597 | + % (b) at (0,1.5) {$b$}; | |
598 | + % % -------------------------------- | |
599 | + % \node [ sub, | |
600 | + % pin=270:\textcolor{blue!50}{$\co{x}$}] | |
601 | + (c) at (4.5,1.5) {$c$}; | |
602 | + % -------------------------------- | |
603 | + \node [ sm, | |
604 | + pin=45:\textcolor{teal}{$1$}] | |
605 | + (A) at (8.5,1.5) {$\co{a}$}; | |
606 | + % -------------------------------- | |
607 | + % \node [ ind, | |
608 | + % pin=270:\textcolor{purple}{$0$}] | |
609 | + % (B) at (9.5, 1.5) {$\co{b}$}; | |
610 | + % % -------------------------------- | |
611 | + % \node [ ind, | |
612 | + % pin=270:\textcolor{purple}{$0$}] | |
613 | + % (C) at (10.5, 1.5) {$\co{c}$}; | |
614 | + % -------------------------------- | |
615 | + % | |
616 | + % -------------------------------- | |
617 | + % \node [ sm, | |
618 | + % pin=90:\textcolor{teal}{$x$}] | |
619 | + % (ab) at (0,4) {$ab$}; | |
620 | + % % -------------------------------- | |
621 | + % \node [ sm, | |
622 | + % pin=90:\textcolor{teal}{$\co{x}$}] | |
623 | + % (ac) at (3,4) {$ac$}; | |
624 | + % % -------------------------------- | |
625 | + % \node [ ind, | |
626 | + % pin=270:\textcolor{purple}{$0$}] | |
627 | + % (aB) at (1,4) {$a\co{b}$}; | |
628 | + % % -------------------------------- | |
629 | + % \node [ ind, | |
630 | + % pin=270:\textcolor{purple}{$0$}] | |
631 | + % (aC) at (2,4) {$a\co{c}$}; | |
632 | + % -------------------------------- | |
633 | + \node [ sup, | |
634 | + pin=135:\textcolor{blue!50}{$1$}] | |
635 | + (Ab) at (4,4) {$\co{a}b$}; | |
636 | + % -------------------------------- | |
637 | + \node [ sup, | |
638 | + pin=135:\textcolor{blue!50}{$1$}] | |
639 | + (Ac) at (5,4) {$\co{a}c$}; | |
640 | + % -------------------------------- | |
641 | + \node [ sup, | |
642 | + pin=135:\textcolor{blue!50}{$1$}] | |
643 | + (AB) at (6,4) {$\co{a}\co{b}$}; | |
644 | + % -------------------------------- | |
645 | + \node [ sup, | |
646 | + pin=135:\textcolor{blue!50}{$1$}] | |
647 | + (AC) at (7,4) {$\co{a}\co{c}$}; | |
648 | + % -------------------------------- | |
649 | + % \node [ ind, | |
650 | + % pin=270:\textcolor{purple}{$0$}] | |
651 | + % (bc) at (10,4) {$bc$}; | |
652 | + % % -------------------------------- | |
653 | + % \node [ ind, | |
654 | + % pin=270:\textcolor{purple}{$0$}] | |
655 | + % (bC) at (11,4) {$b\co{c}$}; | |
656 | + % % -------------------------------- | |
657 | + % \node [ ind, | |
658 | + % pin=270:\textcolor{purple}{$0$}] | |
659 | + % (Bc) at (9.5,3.5) {$\co{b}c$}; | |
660 | + % % -------------------------------- | |
661 | + % \node [ ind, | |
662 | + % pin=270:\textcolor{purple}{$0$}] | |
663 | + % (BC) at (10.5,3.5) {$\co{b}\co{c}$}; | |
664 | + % -------------------------------- | |
665 | + % | |
666 | + % -------------------------------- | |
667 | + % \node [ sup, | |
668 | + % pin=90:\textcolor{blue!75}{$x\co{x}$}] | |
669 | + % (abc) at (1.5,6) | |
670 | + % {$abc$}; | |
671 | + % % -------------------------------- | |
672 | + % \node [ sup, | |
673 | + % pin=90:\textcolor{blue!50}{$x$}] | |
674 | + % (abC) at (0,6) {$ab\co{c}$}; | |
675 | + % % -------------------------------- | |
676 | + % \node [ sup, | |
677 | + % pin=90:\textcolor{blue!50}{$\co{x}$}] | |
678 | + % (aBc) at (3,6) {$a\co{b}c$}; | |
679 | + % % -------------------------------- | |
680 | + % \node [ ind, | |
681 | + % pin=90:\textcolor{purple}{$0$}] | |
682 | + % (aBC) at (5,6) {$a\co{b}\co{c}$}; | |
683 | + % -------------------------------- | |
684 | + \node [ sup, | |
685 | + pin=45:\textcolor{blue!50}{$1$}] | |
686 | + (Abc) at (7,6) {$\co{a}bc$}; | |
687 | + % -------------------------------- | |
688 | + \node [ sup, | |
689 | + pin=45:\textcolor{blue!50}{$1$}] | |
690 | + (AbC) at (8,6) {$\co{a}b\co{c}$}; | |
691 | + % -------------------------------- | |
692 | + \node [ sup, | |
693 | + pin=45:\textcolor{blue!50}{$1$}] | |
694 | + (ABc) at (9,6) {$\co{a}\co{b}c$}; | |
695 | + % -------------------------------- | |
696 | + \node [ sup, | |
697 | + pin=45:\textcolor{blue!50}{$1$}] | |
698 | + (ABC) at (10,6) {$\co{a}\co{b}\co{c}$}; | |
699 | + % -------------------------------- | |
700 | + % | |
701 | + % -------------------------------- | |
702 | + % \draw [->] (ab) to [out=270,in=180] (E); | |
703 | + % \draw [->] (ab) to [out=270,in=90] (a); | |
704 | + % \draw [->] (ab) to [out=270,in=90] (b); | |
705 | + % \draw [->] (ab) to [out=90,in=270] (abc); | |
706 | + % \draw [->] (ab) to [out=90,in=270] (abC); | |
707 | + % % | |
708 | + % \draw [->] (ac) to [out=270,in=180] (E); | |
709 | + % \draw [->] (ac) to [out=270,in=90] (a); | |
710 | + % \draw [->] (ac) to [out=270,in=90] (c); | |
711 | + % \draw [->] (ac) to [out=90,in=270] (abc); | |
712 | + % \draw [->] (ac) to [out=90,in=270] (aBc); | |
384 | 713 | % |
385 | 714 | \draw [->] (A) to [out=270,in=0] (E); |
386 | 715 | % |
... | ... | @@ -415,22 +744,22 @@ rrect/.style={ |
415 | 744 | w_d\at{x} = 0.\label{def:w.inconsistent} |
416 | 745 | \end{equation} |
417 | 746 | % |
418 | - \item $z$ is an interpretation and $\lset{z} = \set{z} = \uset{z}$. Then $z$ is a stable model and \textbf{define} | |
747 | + \item $z$ is an event and $\lset{z} = \set{z} = \uset{z}$. Then $z$ is a stable model and \textbf{define} | |
419 | 748 | \begin{equation} |
420 | 749 | w_d\at{z} = w\at{z} = d\at{z} \pr{\theta_z}.\label{eq:prob.sm} |
421 | 750 | \end{equation} |
422 | 751 | % |
423 | - \item $z$ is an interpretation and $\lset{z} \neq \emptyset \land \uset{x} = \emptyset$. Then \textbf{define} | |
752 | + \item $z$ is an event and $\lset{z} \neq \emptyset \land \uset{x} = \emptyset$. Then \textbf{define} | |
424 | 753 | \begin{equation} |
425 | 754 | w_d\at{z} = \sum_{s \in \lset{z}} w_d\at{s}.\label{def:w.disj} |
426 | 755 | \end{equation} |
427 | 756 | % |
428 | - \item $z$ is an interpretation and $\lset{z} = \emptyset \land \uset{z} \neq \emptyset$. Then \textbf{define} | |
757 | + \item $z$ is an event and $\lset{z} = \emptyset \land \uset{z} \neq \emptyset$. Then \textbf{define} | |
429 | 758 | \begin{equation} |
430 | 759 | w_d\at{z} = \prod_{s \in \uset{z}} w_d\at{s}.\label{def:w.conj} |
431 | 760 | \end{equation} |
432 | 761 | % |
433 | - \item $z$ is an interpretation and $\lset{z} = \emptyset \land \uset{z} = \emptyset$. Then \textbf{define} | |
762 | + \item $z$ is an event and $\lset{z} = \emptyset \land \uset{z} = \emptyset$. Then \textbf{define} | |
434 | 763 | \begin{equation} |
435 | 764 | w_d\at{z} = 0.\label{def:w.empty} |
436 | 765 | \end{equation} |
... | ... | @@ -442,7 +771,7 @@ rrect/.style={ |
442 | 771 | % |
443 | 772 | \item The last point defines a ``weight'' function on the samples that depends not only on the total choices and stable models of a PASP but also on a certain function $d$ that must respect some conditions. To simplify the notation we use the subscript in $w_d$ only when necessary. |
444 | 773 | % |
445 | - \item At first, it may seem counter-intuitive that $w\at{\emptyset} = \sum_{s\in\fml{S}} w\at{s}$ is the largest ``weight'' in the lattice. But $\emptyset$, as an interpretation, sets zero restrictions on the ``compatible'' stable models. The ``complement'' of $\bot = \emptyset$ is the \emph{maximal inconsistent} sample $\top = \fml{A} \cup \cset{\neg a }{ a \in \fml{A}}$. | |
774 | + \item At first, it may seem counter-intuitive that $w\at{\emptyset} = \sum_{s\in\fml{S}} w\at{s}$ is the largest ``weight'' in the lattice. But $\emptyset$, as an event, sets zero restrictions on the ``compatible'' stable models. The ``complement'' of $\bot = \emptyset$ is the \emph{maximal inconsistent} sample $\top = \fml{A} \cup \cset{\neg a }{ a \in \fml{A}}$. | |
446 | 775 | % |
447 | 776 | \item \textbf{We haven't yet defined a probability measure.} To do so we must define a set of samples $\Omega$, a set of events $F\subseteq \pset{\Omega}$ and a function $P:F\to\intcc{0,1}$ such that: |
448 | 777 | \begin{enumerate} |
... | ... | @@ -511,12 +840,12 @@ rrect/.style={ |
511 | 840 | \end{frame} |
512 | 841 | % ================================================================ |
513 | 842 | \begin{frame} |
514 | - Suppose that we add an annotation $\alpha :: a$, which entails $\co{\alpha} :: \neg a$. This is enough to get $w\at{s_1} = \co{\alpha}$ but, on the absence of further information, no fixed probability can be assigned to either model $s_2, s_3$ except that the respective sum must be $\alpha$. So, expressing our lack of knowledge using a parameter $d \in \intcc{0, 1}$ we get: | |
843 | + Suppose that we add an annotation $x :: a$, which entails $\co{x} :: \neg a$. This is enough to get $w\at{s_1} = \co{x}$ but, on the absence of further information, no fixed probability can be assigned to either model $s_2, s_3$ except that the respective sum must be $x$. So, expressing our lack of knowledge using a parameter $d \in \intcc{0, 1}$ we get: | |
515 | 844 | $$ |
516 | 845 | \begin{cases} |
517 | - w\at{s_1 } = &\co{\alpha}\cr | |
518 | - w\at{s_2 } = &d\alpha\cr | |
519 | - w\at{s_3} = &\co{d}\alpha. | |
846 | + w\at{s_1 } = &\co{x}\cr | |
847 | + w\at{s_2 } = &dx\cr | |
848 | + w\at{s_3} = &\co{d}x. | |
520 | 849 | \end{cases} |
521 | 850 | $$ |
522 | 851 | \end{frame} |
... | ... | @@ -527,19 +856,19 @@ rrect/.style={ |
527 | 856 | \begin{itemize} |
528 | 857 | \item Negations are represented as \emph{e.g.} $\co{a}$ instead of $\neg a$; Stable models are denoted by shaded nodes as \tikz{\node[fill=gray!50] {$ab$}}. |
529 | 858 | |
530 | - \item Interpretations in $\lset{x}$ are \emph{e.g.} \tikz{\node[draw, circle] {$a$}} and those in $\uset{x}$ are \emph{e.g.} \tikz{\node[draw] {$\co{a}b$}}. The remaining are simply denoted by \emph{e.g.} \tikz{\node {$a\co{b}$}}. | |
859 | + \item Events in $\lset{x}$ are \emph{e.g.} \tikz{\node[ circle] {$a$}} and those in $\uset{x}$ are \emph{e.g.} \tikz{\node[draw] {$\co{a}b$}}. The remaining are simply denoted by \emph{e.g.} \tikz{\node {$a\co{b}$}}. | |
531 | 860 | |
532 | - \item The edges connect stable models with related interpretations. Up arrow indicate links to $\uset{s}$ and down arrows to $\lset{s}$. | |
861 | + \item The edges connect stable models with related events. Up arrow indicate links to $\uset{s}$ and down arrows to $\lset{s}$. | |
533 | 862 | |
534 | 863 | \item The \emph{weight propagation} sets: |
535 | 864 | $$ |
536 | 865 | \begin{aligned} |
537 | - w\at{abc} &= w\at{ab} w\at{ac} = \alpha^2d\co{d}, \cr | |
538 | - w\at{\co{a}\cdot\cdot} &= w\at{\neg a} = \co{\alpha}, \cr | |
539 | - w\at{a} &= w\at{ab} + w\at{ac} = \alpha(d + \co{d}) = \alpha, \cr | |
540 | - w\at{b} &= w\at{ab} = d\alpha, \cr | |
541 | - w\at{c} &= w\at{ac} = \co{d}\alpha, \cr | |
542 | - w\at{\emptyset} &= w\at{ab} + w\at{ac} + w\at{\neg a} = d\alpha + \co{d}\alpha + \co{\alpha} = 1, \cr | |
866 | + w\at{abc} &= w\at{ab} w\at{ac} = x^2d\co{d}, \cr | |
867 | + w\at{\co{a}\cdot\cdot} &= w\at{\neg a} = \co{x}, \cr | |
868 | + w\at{a} &= w\at{ab} + w\at{ac} = x(d + \co{d}) = x, \cr | |
869 | + w\at{b} &= w\at{ab} = dx, \cr | |
870 | + w\at{c} &= w\at{ac} = \co{d}x, \cr | |
871 | + w\at{\emptyset} &= w\at{ab} + w\at{ac} + w\at{\neg a} = dx + \co{d}x + \co{x} = 1, \cr | |
543 | 872 | w\at{a\co{b}} &= 0. |
544 | 873 | \end{aligned} |
545 | 874 | $$ |
... | ... | @@ -551,16 +880,16 @@ rrect/.style={ |
551 | 880 | &+ w\at{a}+ w\at{b}+ w\at{c}\cr |
552 | 881 | &+ w\at{\emptyset}\cr |
553 | 882 | % |
554 | - &= - \alpha^{2} d^{2} + \alpha^{2} d + 2 \alpha d - 7 \alpha + 10 | |
883 | + &= - x^{2} d^{2} + x^{2} d + 2 x d - 7 x + 10 | |
555 | 884 | \end{aligned} |
556 | 885 | $$ |
557 | - \item Now, if $\alpha$ has an annotation to \emph{e.g.} $0.3$ we get | |
886 | + \item Now, if $x$ has an annotation to \emph{e.g.} $0.3$ we get | |
558 | 887 | $$ |
559 | 888 | Z = - 0.09 d^{2} + 0.69 d + 7.9 |
560 | 889 | $$ |
561 | 890 | \item Now some statistics are possible. For example we get |
562 | 891 | $$ |
563 | - \pr{abc \mid \alpha = 0.3} = \frac{0.09 d \left(d - 1\right)}{0.09 d^{2} - 0.69 d - 7.9} | |
892 | + \pr{abc \mid x = 0.3} = \frac{0.09 d \left(d - 1\right)}{0.09 d^{2} - 0.69 d - 7.9} | |
564 | 893 | $$. |
565 | 894 | |
566 | 895 | \item This expression can be plotted for $d\in\intcc{0,1}$ |
... | ... | @@ -571,7 +900,7 @@ rrect/.style={ |
571 | 900 | \item If a data set $E$ entails \emph{e.g.} $\pr{abc \mid E} = 0.0015$ we can numerically solve |
572 | 901 | $$ |
573 | 902 | \begin{aligned} |
574 | - \pr{abc \mid \alpha = 0.3} &= \pr{abc \mid E} \cr | |
903 | + \pr{abc \mid x = 0.3} &= \pr{abc \mid E} \cr | |
575 | 904 | \iff\cr |
576 | 905 | \frac{0.09 d \del{d - 1}}{0.09 d^{2} - 0.69 d - 7.9} &= 0.0015 |
577 | 906 | \end{aligned} |
... | ... | @@ -613,9 +942,9 @@ rrect/.style={ |
613 | 942 | % ================================================================ |
614 | 943 | \begin{frame} |
615 | 944 | |
616 | - Looking into probabilistic interpretations of the program and/or its models, we define $\alpha = \pr{\Theta = \theta_1}\in\intcc{0, 1}$ and $\pr{\Theta = \theta_2} = \co{\alpha}$. | |
945 | + Looking into probabilistic events of the program and/or its models, we define $x = \pr{\Theta = \theta_1}\in\intcc{0, 1}$ and $\pr{\Theta = \theta_2} = \co{x}$. | |
617 | 946 | |
618 | - Since $s_1$ is the only stable model that results from $\Theta = \theta_1$, it is natural to extend $\pr{ s_1 } = \pr{\Theta = \theta_1} = \alpha$. However, there is no clear way to assign $\pr{s_2}, \pr{s_3}$ since \emph{both models result from the single total choice} $\Theta = \theta_2$. Clearly, | |
947 | + Since $s_1$ is the only stable model that results from $\Theta = \theta_1$, it is natural to extend $\pr{ s_1 } = \pr{\Theta = \theta_1} = x$. However, there is no clear way to assign $\pr{s_2}, \pr{s_3}$ since \emph{both models result from the single total choice} $\Theta = \theta_2$. Clearly, | |
619 | 948 | $$\pr{s_2 \mid \Theta} + \pr{s_3 \mid \Theta} = |
620 | 949 | \begin{cases} |
621 | 950 | 0 & \text{if}~\Theta = \theta_1\cr |
... | ... | @@ -639,13 +968,13 @@ rrect/.style={ |
639 | 968 | \begin{array}{cc|l} |
640 | 969 | A, B, C& P & \text{Obs.}\cr |
641 | 970 | \hline |
642 | - a, \neg b, c & \alpha & s_1, \Theta=\theta_1\cr | |
643 | - \neg a, b, \neg c & \co{\alpha}\beta & s_2, \Theta=\theta_2\cr | |
644 | - \neg a, \neg b, c & \co{\alpha}\co{\beta} & s_3, \Theta=\theta_2\cr | |
971 | + a, \neg b, c & x & s_1, \Theta=\theta_1\cr | |
972 | + \neg a, b, \neg c & \co{x}\beta & s_2, \Theta=\theta_2\cr | |
973 | + \neg a, \neg b, c & \co{x}\co{\beta} & s_3, \Theta=\theta_2\cr | |
645 | 974 | \ast & 0&\text{not stable models} |
646 | 975 | \end{array} |
647 | 976 | $$ |
648 | - where $\alpha, \beta\in\intcc{0,1}$. | |
977 | + where $x, \beta\in\intcc{0,1}$. | |
649 | 978 | \end{frame} |
650 | 979 | % ================================================================ |
651 | 980 | \section{Conclusions} |
... | ... | @@ -680,14 +1009,14 @@ rrect/.style={ |
680 | 1009 | |
681 | 1010 | \item The \deft{Herbrand base} of a program is the set of ground literals that result from combining all the predicates and constants of the program. |
682 | 1011 | \begin{itemize} |
683 | - \item An \deft{interpretation} is a consistent subset (\emph{i.e.} doesn't contain $\set{a, \neg a}$) of the Herbrand base. | |
684 | - \item Given an interpretation $I$, a ground literal $a$ is \deft{true}, $I \models a$, if $a \in I$; otherwise the literal is \deft{false}. | |
1012 | + \item An \deft{event} is a consistent subset (\emph{i.e.} doesn't contain $\set{a, \neg a}$) of the Herbrand base. | |
1013 | + \item Given an event $I$, a ground literal $a$ is \deft{true}, $I \models a$, if $a \in I$; otherwise the literal is \deft{false}. | |
685 | 1014 | \item A ground subgoal, $\naf b$, where $b$ is a ground literal, is \deft{true}, $I \models \naf b$, if $b \not\in I$; otherwise, if $b \in I$, it is \deft{false}. |
686 | - \item A ground rule $r = h_1 \vee \cdots \vee h_m \leftarrow b_1 \wedge \cdots \wedge b_n$ is \deft{satisfied} by the interpretation $I$, \emph{i.e.} $I \models r$, iff | |
1015 | + \item A ground rule $r = h_1 \vee \cdots \vee h_m \leftarrow b_1 \wedge \cdots \wedge b_n$ is \deft{satisfied} by the event $I$, \emph{i.e.} $I \models r$, iff | |
687 | 1016 | $$ |
688 | 1017 | \forall j \exists i~I \models b_j \implies I \models h_i. |
689 | 1018 | $$ |
690 | - \item A \deft{model} of a program is an interpretation that satisfies all its rules. Denote $\fml{M}_P$ the set of all models of $P$. | |
1019 | + \item A \deft{model} of a program is an event that satisfies all its rules. Denote $\fml{M}_P$ the set of all models of $P$. | |
691 | 1020 | \end{itemize} |
692 | 1021 | |
693 | 1022 | \item The \deft{dependency graph} of a program is a digraph where: |
... | ... | @@ -701,16 +1030,16 @@ rrect/.style={ |
701 | 1030 | \end{itemize} |
702 | 1031 | \item Every \emph{definite program} has a unique minimal model: its \deft{semantic}. |
703 | 1032 | \item Programs with negation may have no unique minimal model. |
704 | - \item Given a program $P$ and an interpretation $I$, their \deft{reduct}, $P^I$, is the propositional program that results from | |
1033 | + \item Given a program $P$ and an event $I$, their \deft{reduct}, $P^I$, is the propositional program that results from | |
705 | 1034 | \begin{enumerate} |
706 | 1035 | \item Removing all the rules with $\naf b$ in the body where $b \in I$. |
707 | 1036 | \item Removing all the $\naf b$ subgoals from the remaining rules. |
708 | 1037 | \end{enumerate} |
709 | - \item A \deft{stable model} (or \deft{answer set}) of the program $P$ is an interpretation $I$ that is the minimal model of the reduct $P^I$. | |
1038 | + \item A \deft{stable model} (or \deft{answer set}) of the program $P$ is an event $I$ that is the minimal model of the reduct $P^I$. | |
710 | 1039 | \item Denote $\fml{S}_P$ the set of all stable models of program $P$. The \deft{semantics} (or \deft{answer sets}) of a program $P$ is the set $\fml{S}_P$. |
711 | 1040 | \begin{itemize} |
712 | 1041 | \item Some programs, such as $a \leftarrow \naf a$, have no stable models. |
713 | - \item A stable model is an interpretation closed under the rules of the program. | |
1042 | + \item A stable model is an event closed under the rules of the program. | |
714 | 1043 | \end{itemize} |
715 | 1044 | \end{itemize} |
716 | 1045 | \end{frame} | ... | ... |
text/00_PASP.toc
... | ... | @@ -1,6 +0,0 @@ |
1 | -\beamer@sectionintoc {1}{Introduction}{2}{0}{1} | |
2 | -\beamer@sectionintoc {2}{Extending Probability to Samples}{11}{0}{2} | |
3 | -\beamer@sectionintoc {3}{Cases \& Examples}{18}{0}{3} | |
4 | -\beamer@subsectionintoc {3}{1}{Programs with disjunctive heads}{19}{0}{3} | |
5 | -\beamer@subsectionintoc {3}{2}{Non-stratified programs}{23}{0}{3} | |
6 | -\beamer@sectionintoc {4}{Conclusions}{28}{0}{4} |
text/00_PASP.xdv
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1 | +\documentclass[a4paper]{article} | |
2 | +\usepackage{hyperref} | |
3 | +\hypersetup{ | |
4 | + colorlinks=true, | |
5 | + linkcolor=blue, | |
6 | +} | |
7 | +\usepackage{commath} | |
8 | +\usepackage{amssymb} | |
9 | +% | |
10 | +% Local commands | |
11 | +% | |
12 | +\newcommand{\todo}[1]{{\color{orange}TODO #1}} | |
13 | +\newcommand{\naf}{\ensuremath{\sim\!}} | |
14 | +\newcommand{\larr}{\ensuremath{\leftarrow}} | |
15 | +\newcommand{\at}[1]{\ensuremath{\!\del{#1}}} | |
16 | +\newcommand{\co}[1]{\ensuremath{\overline{#1}}} | |
17 | +\newcommand{\fml}[1]{\ensuremath{{\cal #1}}} | |
18 | +\newcommand{\deft}[1]{\textbf{#1}} | |
19 | +\newcommand{\pset}[1]{\ensuremath{\mathbb{P}\at{#1}}} | |
20 | +\newcommand{\ent}{\ensuremath{\lhd}} | |
21 | +\newcommand{\cset}[2]{\ensuremath{\set{#1,~#2}}} | |
22 | +\newcommand{\langof}[1]{\ensuremath{\fml{L}\at{#1}}} | |
23 | +\newcommand{\uset}[1]{\ensuremath{\left|{#1}\right>}} | |
24 | +\newcommand{\lset}[1]{\ensuremath{\left<{#1}\right|}} | |
25 | +\newcommand{\pr}[1]{\ensuremath{\mathrm{p}\at{#1}}} | |
26 | +\newcommand{\given}{\ensuremath{~\middle|~}} | |
27 | + | |
28 | +\title{Zugzwang\\\textit{Logic and Artificial Intelligence}} | |
29 | +\author{Francisco Coelho\\ \texttt{fc@uevora.pt}} | |
30 | + | |
31 | +\begin{document} | |
32 | +\maketitle | |
33 | + | |
34 | +\begin{abstract} | |
35 | + A major limitation of logical representations is the implicit assumption that the Background Knowledge (BK) is perfect. This assumption is problematic if data is noisy, which is often the case. Here we aim to explore how ASP specifications with probabilistic facts can lead to characterizations of probability functions on the specification's domain. | |
36 | +\end{abstract} | |
37 | + | |
38 | +\section{Introduction and Motivation } | |
39 | + | |
40 | +Answer Set Programming (ASP) is a logic programming paradigm based on the Stable Model semantics of Normal Logic Programs (NP) that can be implemented using the latest advances in SAT solving technology. ASP is a truly declarative language that supports language constructs such as disjunction in the head of a clause, choice rules, and hard and weak constraints. | |
41 | + | |
42 | +The Distribution Semantics (DS) is a key approach to extend logical representations with probabilistic reasoning. Probabilistic Facts (PF) are the most basic stochastic DS primitive and they take the form of logical facts, $a$, labelled with a probability, such as $p::a$; Each probabilistic fact represents a boolean random variable that is true with probability $p$ and false with probability $1 - p$. A (consistent) combination of the PFs defines a \textit{total choice} $c = \set{p::a, \ldots}$ such that | |
43 | + | |
44 | +\begin{equation} | |
45 | + P(c) = \prod_{a\in c} p \prod_{a \not\in c} (1- p). | |
46 | +\end{equation} | |
47 | + | |
48 | +Our goal is to extend this probability, from total choices, to cover the specification domain. We can foresee two key applications of this extended probability: | |
49 | + | |
50 | +\begin{enumerate} | |
51 | + \item Support any probabilistic reasoning/task on the specification domain. | |
52 | + \item Also, given a dataset and a divergence measure, now the specification can be scored (by the divergence w.r.t. the \emph{empiric} distribution of the dataset), and sorted amongst other specifications. This is a key ingredient in algorithms searching, for example, an \textit{optimal specification} of the dataset. | |
53 | +\end{enumerate} | |
54 | + | |
55 | +This goal faces a critical problem concerning situations where \textit{multiple} standard models result from a given total choice, illustrated by the following example. The specification | |
56 | +$$ | |
57 | +\begin{aligned} | |
58 | + 0.3::a&,\cr | |
59 | + b \vee c& \leftarrow a. | |
60 | +\end{aligned} | |
61 | +$$ | |
62 | +has three stable models, $\set{\neg a}, \set{a, b}$ and $\set{a, c}$. While it is straightforward to set $P(\neg a)=0.7$, there is \textit{no further information} to assign values to $P(a,b)$ and $P(a,c)$. At best, we can use a parameter $\alpha$ such that | |
63 | +$$ | |
64 | +\begin{aligned} | |
65 | +P(a,b) &= 0.3 \alpha,\cr | |
66 | +P(a,c) &= 0.3 (1 - \alpha). | |
67 | +\end{aligned} | |
68 | +$$ | |
69 | + | |
70 | +This uncertainty in inherent to the specification, but can be mitigated with the help of a dataset: the parameter $\alpha$ can be estimated from the empirical distribution. | |
71 | + | |
72 | +In summary, if an ASP specification is intended to describe some system that can be observed then: | |
73 | + | |
74 | +\begin{enumerate} | |
75 | + \item The observations can be used to estimate the value of the parameters (such as $\alpha$ above and others entailed from further clauses). | |
76 | + \item With a probability set for the stable models, we want to extend it to all the samples (\textit{i.e.} consistent sets of literals) of the specification. | |
77 | + \item This extended probability can then be related to the \textit{empirical distribution}, using a probability divergence, such as Kullback-Leibler; and the divergence value used as a \textit{performance} measure of the specification with respect to the observations. | |
78 | + \item If that specification is only but one of many possible candidates then that performance measure can be used, \textit{e.g.} as fitness, by algorithms searching (optimal) specifications of a dataset of observations. | |
79 | +\end{enumerate} | |
80 | + | |
81 | +Currently, we are on the step two above: Extending a probability function (with parameters such as $\alpha$), defined on the stable sets of a specification, to all the samples of the specification. This extension must, of course, respect the axioms of probability so that probabilistic reasoning is consistent with the ASP specification. | |
82 | + | |
83 | +\section{Extending Probabilities} | |
84 | + | |
85 | +Given an ASP specification, we look at the \textit{total choices} $\fml{C}$, \textit{stable models} $\fml{S}$, \textit{atoms} and \textit{literals} $\fml{A}, \fml{L}$, \textit{samples} $z \in \fml{Z} \iff z \subseteq \fml{L}$ and \textit{events} $\fml{E}$ (consistent samples). | |
86 | + | |
87 | +\begin{enumerate} | |
88 | + \item \textbf{Given} $\pr{C = c}$ from the specification. Each total choice $C = c$ (together with the specification facts and rules) entails some stable models, $S_c$. | |
89 | + \item \textbf{Given} $\pr{S = s \given C = c}$ such that its value is zero if $s \not\in S_c$. | |
90 | + \item Each event $E = e$ either: | |
91 | + \begin{enumerate} | |
92 | + \item Is a stable model. Then | |
93 | + \begin{equation} | |
94 | + \pr{E = e \given C = c} = \pr{S = e \given C = c}. | |
95 | + \end{equation} | |
96 | + \item \textit{Is contained} in some stable models. Then | |
97 | + \begin{equation} | |
98 | + \pr{E = e \given C = c} = \sum_{s \supseteq e}\pr{S = s \given C = c}. | |
99 | + \end{equation} | |
100 | + \item \textit{Contains} some stable models. Then | |
101 | + \begin{equation} | |
102 | + \pr{E = e \given C = c} = \prod_{s \subseteq e}\pr{S = s \given C = c}. | |
103 | + \end{equation} | |
104 | + \item Neither contains nor is contained by a stable model. Then | |
105 | + \begin{equation} | |
106 | + \pr{E = e \given C = c} = 0. | |
107 | + \end{equation} | |
108 | + \end{enumerate} | |
109 | + \item For each sample $Z = z$ | |
110 | + \begin{equation} | |
111 | + \pr{Z = z \given C = c} = \begin{cases} | |
112 | + \pr{E = z \given C = c} & z \in \fml{E}, \cr | |
113 | + 0 & \text{otherwise}. | |
114 | + \end{cases} | |
115 | + \end{equation} | |
116 | +\end{enumerate} | |
117 | + | |
118 | +\begin{quotation}\marginpar{Todo} Prove the four event cases, and support the sum/product options, with the independence assumptions. | |
119 | +\end{quotation} | |
120 | + | |
121 | + | |
122 | + | |
123 | +\section*{References} | |
124 | + | |
125 | + | |
126 | +\begin{enumerate} | |
127 | + \item Victor Verreet, Vincent Derkinderen, Pedro Zuidberg Dos Martires, Luc De Raedt, Inference and Learning with Model Uncertainty in Probabilistic Logic Programs (2022) | |
128 | + \item Andrew Cropper, Sebastijan Dumancic, Richard Evans, Stephen H. Muggleton, Inductive logic programming at 30 (2021) | |
129 | + \item Fabio Gagliardi Cozman, Denis Deratani Mauá, The joy of Probabilistic Answer Set Programming: Semantics - complexity, expressivity, inference (2020) | |
130 | + \item Fabrizio Riguzzi, Foundations of Probabilistic Logic Programming Languages, Semantics, Inference and Learning. Rivers Publishers (2018) | |
131 | + \item Martin Gebser, Roland Kaminski, Benjamin Kaufmann, and Torsten Schaub, Answer Set Solving in Practice, Morgan \& Claypool Publishers (2013) | |
132 | +\end{enumerate} | |
133 | + | |
134 | +\end{document} | |
0 | 135 | \ No newline at end of file | ... | ... |